In a previous job with the American Diabetes Association, I was part of a project to build Diabetes PHD - Personal Health Decisions. It's an online health profiler that takes information about you - your height, weight, medical history, etc. - and runs a series of complex mathematical equations to determine your risk for certain health complications like stroke and heart disease. The math is very complex and requires a lot of computing power. It would take one very powerful computer hours if not days to return the results.
To increase the response time, Diabetes PHD uses the power of distributed computing. In the simplest terms, this is where large math problems are broken up into small problems, computed separately using multiple computers and then reassembled to get the answer. The math work for Diabetes PHD is sent out onto a grid of hundreds of computers working simultaneously and is able to return results in just a few minutes.
At any particular time, your computer is not working all that hard. Maybe you're just editing a spreadsheet or working on a Word document. Or you're asleep but your computer is still on. Distributed computing is a way to use some of each computer's unused processing power. The application often runs as a screensaver or continuously runs in the background while a user works.
Many other interesting projects use the power of distributed computing. The most famous may be the SETI project - the Search for ExtraTerrestrial Intelligence. You can find a full list of projects here and even find out how to sign up to donate the unused processing power of your computer.